Embedded newform 76.5.g.a.7.18

• Label: 76.5.g.a.7.18
• Level: $76$
• Weight: $5$
• Character: 76.7
• Analytic conductor: $7.856$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85611719437\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.18
Character	\(\chi\)	\(=\)	76.7
Dual form			76.5.g.a.11.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.381196 - 3.98179i) q^{2} +(-12.2124 + 7.05085i) q^{3} +(-15.7094 + 3.03569i) q^{4} +(-5.66259 - 9.80789i) q^{5} +(32.7304 + 45.9397i) q^{6} +30.6646i q^{7} +(18.0759 + 61.3943i) q^{8} +(58.9291 - 102.068i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - q^{2} + 13 q^{4} - 2 q^{5} - 15 q^{6} - 94 q^{8} + 972 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.381196	−	3.98179i	−0.0952991	−	0.995449i
							\(3\)	−12.2124	+	7.05085i	−1.35694	+	0.783428i	−0.989210	−	0.146505i	\(-0.953197\pi\)
−0.367727	+	0.929934i	\(0.619864\pi\)
\(5\)	−5.66259	−	9.80789i	−0.226504	−	0.392316i	0.730266	−	0.683163i	\(-0.239396\pi\)
							−0.956769	+	0.290847i	\(0.906063\pi\)
\(7\)			30.6646i			0.625808i	0.949785	+	0.312904i	\(0.101302\pi\)
							−0.949785	+	0.312904i	\(0.898698\pi\)
\(11\)		−	56.8189i		−	0.469578i	−0.972046	−	0.234789i	\(-0.924560\pi\)
							0.972046	−	0.234789i	\(-0.0754399\pi\)
\(13\)	106.903	−	185.162i	0.632563	−	1.09563i	−0.354463	−	0.935070i	\(-0.615336\pi\)
							0.987026	−	0.160561i	\(-0.0513303\pi\)
\(17\)	33.3271	+	57.7243i	0.115319	+	0.199738i	0.917907	−	0.396795i	\(-0.129878\pi\)
							−0.802588	+	0.596533i	\(0.796544\pi\)
\(19\)	268.028	+	241.831i	0.742459	+	0.669892i
							\(23\)	395.760	+	228.492i	0.748128	+	0.431932i	0.825017	−	0.565108i	\(-0.191165\pi\)
−0.0768891	+	0.997040i	\(0.524499\pi\)
\(29\)	−534.019	+	924.948i	−0.634981	+	1.09982i	0.351538	+	0.936173i	\(0.385659\pi\)
							−0.986519	+	0.163646i	\(0.947675\pi\)
\(31\)			91.5075i			0.0952211i	0.998866	+	0.0476106i	\(0.0151606\pi\)
							−0.998866	+	0.0476106i	\(0.984839\pi\)
\(37\)	1289.56			0.941973			0.470987	−	0.882140i	\(-0.343898\pi\)
							0.470987	+	0.882140i	\(0.343898\pi\)
\(41\)	1170.66	+	2027.64i	0.696405	+	1.20621i	0.969705	+	0.244280i	\(0.0785516\pi\)
							−0.273300	+	0.961929i	\(0.588115\pi\)
\(43\)	577.756	−	333.567i	0.312469	−	0.180404i	−0.335562	−	0.942018i	\(-0.608926\pi\)
							0.648031	+	0.761614i	\(0.275593\pi\)
\(47\)	−1817.43	−	1049.30i	−0.822741	−	0.475010i	0.0286199	−	0.999590i	\(-0.490889\pi\)
							−0.851361	+	0.524581i	\(0.824222\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.g.a.7.18	yes	76
4.3	odd	2	inner	76.5.g.a.7.8	✓	76
19.11	even	3	inner	76.5.g.a.11.8	yes	76
76.11	odd	6	inner	76.5.g.a.11.18	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.g.a.7.8	✓	76	4.3	odd	2	inner
76.5.g.a.7.18	yes	76	1.1	even	1	trivial
76.5.g.a.11.8	yes	76	19.11	even	3	inner
76.5.g.a.11.18	yes	76	76.11	odd	6	inner